Algebra Examples

$\sqrt{(1 - \csc^{2} (y)) (\cos^{2} (y) - 1)}$

Step 1

Rewrite $1$ as $1^{2}$ .

$\sqrt{(1^{2} - \csc^{2} (y)) (\cos^{2} (y) - 1)}$

Step 2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \csc (y)$ .

$\sqrt{(1 + \csc (y)) (1 - \csc (y)) (\cos^{2} (y) - 1)}$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Rewrite $\csc (y)$ in terms of sines and cosines.

$\sqrt{(1 + \frac{1}{\sin (y)}) (1 - \csc (y)) (\cos^{2} (y) - 1)}$

Step 3.2

Rewrite $\csc (y)$ in terms of sines and cosines.

$\sqrt{(1 + \frac{1}{\sin (y)}) (1 - \frac{1}{\sin (y)}) (\cos^{2} (y) - 1)}$

Step 4

Rewrite $1$ as $1^{2}$ .

$\sqrt{(1 + \frac{1}{\sin (y)}) (1 - \frac{1}{\sin (y)}) (\cos^{2} (y) - 1^{2})}$

Step 5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \cos (y)$ and $b = 1$ .

$\sqrt{(1 + \frac{1}{\sin (y)}) (1 - \frac{1}{\sin (y)}) (\cos (y) + 1) (\cos (y) - 1)}$

Step 6

Convert from $\frac{1}{\sin (y)}$ to $\csc (y)$ .

$\sqrt{(1 + \frac{1}{\sin (y)}) (1 - \csc (y)) (\cos (y) + 1) (\cos (y) - 1)}$

Step 7

Convert from $\frac{1}{\sin (y)}$ to $\csc (y)$ .

$\sqrt{(1 + \csc (y)) (1 - \csc (y)) (\cos (y) + 1) (\cos (y) - 1)}$